<h2>Problem 295</h2>
<div style="color:#666;font-size:80%;">05 June 2010</div><br />
<div class="problem_content">
<p>We call the convex area enclosed by two circles a <i>lenticular hole</i> if:
<ul>
<li>The centres of both circles are on lattice points.</li>
<li>The two circles intersect at two distinct lattice points.</li>
<li>The interior of the convex area enclosed by both circles does not contain any lattice points.
</li>
</ul>
</p>
<p>Consider the circles:<BR />
C<img src="" style="display:none;" alt="_(" /><sub>0</sub><img src="" style="display:none;" alt=")" />: <var>x</var><img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" />+<var>y</var><img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" />=25<BR />
C<img src="" style="display:none;" alt="_(" /><sub>1</sub><img src="" style="display:none;" alt=")" />: (<var>x</var>+4)<img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" />+(<var>y</var>-4)<img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" />=1<BR />
C<img src="" style="display:none;" alt="_(" /><sub>2</sub><img src="" style="display:none;" alt=")" />: (<var>x</var>-12)<img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" />+(<var>y</var>-4)<img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" />=65
</p>
<p>
The circles C<img src="" style="display:none;" alt="_(" /><sub>0</sub><img src="" style="display:none;" alt=")" />, C<img src="" style="display:none;" alt="_(" /><sub>1</sub><img src="" style="display:none;" alt=")" /> and C<img src="" style="display:none;" alt="_(" /><sub>2</sub><img src="" style="display:none;" alt=")" /> are drawn in the picture below.</p>
<div align='center'><img src="project/images/p295_lenticular.gif" /></div>
<p>
C<img src="" style="display:none;" alt="_(" /><sub>0</sub><img src="" style="display:none;" alt=")" /> and C<img src="" style="display:none;" alt="_(" /><sub>1</sub><img src="" style="display:none;" alt=")" /> form a lenticular hole, as well as C<img src="" style="display:none;" alt="_(" /><sub>0</sub><img src="" style="display:none;" alt=")" /> and C<img src="" style="display:none;" alt="_(" /><sub>2</sub><img src="" style="display:none;" alt=")" />.</p>
<p>
We call an ordered pair of positive real numbers (r<img src="" style="display:none;" alt="_(" /><sub>1</sub><img src="" style="display:none;" alt=")" />, r<img src="" style="display:none;" alt="_(" /><sub>2</sub><img src="" style="display:none;" alt=")" />) a <i>lenticular pair</i> if there exist two circles with radii r<img src="" style="display:none;" alt="_(" /><sub>1</sub><img src="" style="display:none;" alt=")" /> and r<img src="" style="display:none;" alt="_(" /><sub>2</sub><img src="" style="display:none;" alt=")" /> that form a lenticular hole.
We can verify that (1, 5) and (5, <img src='images/symbol_radic.gif' width='14' height='16' alt='&radic;' border='0' style='vertical-align:middle;' />65) are the lenticular pairs of the example above.</p>
<p>
Let L(N) be the number of <b>distinct</b> lenticular pairs (r<img src="" style="display:none;" alt="_(" /><sub>1</sub><img src="" style="display:none;" alt=")" />, r<img src="" style="display:none;" alt="_(" /><sub>2</sub><img src="" style="display:none;" alt=")" />) for which 0 <img src='images/symbol_lt.gif' width='10' height='10' alt='&lt;' border='0' style='vertical-align:middle;' /> r<img src="" style="display:none;" alt="_(" /><sub>1</sub><img src="" style="display:none;" alt=")" /> <img src='images/symbol_le.gif' width='10' height='12' alt='&le;' border='0' style='vertical-align:middle;' /> r<img src="" style="display:none;" alt="_(" /><sub>2</sub><img src="" style="display:none;" alt=")" /> <img src='images/symbol_le.gif' width='10' height='12' alt='&le;' border='0' style='vertical-align:middle;' /> N.<br />
We can verify that L(10) = 30 and L(100) = 3442.</p>
<p>
Find L(100 000).
</p>












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